Explicative note to this paper - McGill Physicsgang/eprints/eprintLovejoy/neweprint/climate... ·...

Explicative note to this paper 1

This paper was submitted to BAMS on June 27, 2012, it was accepted for 2 publication by editor C. Deser on Dec. 27, 2012 and retracted by the senior BAMS editor 3 J. Rosenfeld on Oct. 24, 2013. The reason given was that a much shorter review of 4 similar material (29% of the length) had been published in the weekly magazine EOS 5 (Lovejoy, S., 2013: What is Climate?: EOS, 94, No. 1, 1 January 2013, p1-2.). Lovejoy 6 was therefore accused of attempting “concurrent dual publication”. 7

The background was that at the moment of submission, the draft paper was posted 8 on Lovejoy’s personal web site. It rapidly gained notoriety in the blogosphere and came 9 to the attention of the EOS editors R. Pielke Sr. and B. Richman. In August 2012, this 10 much shorter review was explicitly commissioned by EOS to cover similar material but 11 in much shorter form and for a quite a different audience. The practice of EOS 12 commissioning short reviews of new work covered at greater length in other journals is 13 common and there was certainly no attempt at dual publication. More details on this 14 unfortunate episode may be found at in the chronology document 15 (http://www.physics.mcgill.ca/~gang/eprints/eprintLovejoy/neweprint/ChronologyBAMS16 .EOS.25.10.13.pdf). 17 18

19

The climate is not what you expect 20

S. Lovejoy1, D. Schertzer2 21

1Physics, McGill University 22

3600 University st., 23

Montreal, Que., Canada 24

Email: [email protected] 25

26

2CEREVE 27

Université Paris Est, 28

6-‐8, avenue Blaise Pascal 29

Cité Descartes 30

77455 MARNE-‐LA-‐VALLEE Cedex, France 31

32

The climate is not what you expect 33

Capsule Summary: 34

Contrary to popular ideas about the climate, what you really expect is 35

macroweather. On scales of a human generation, the climate is what you get. 36

Abstract: 37

Prevailing definitions of climate are not much different from “the climate is 38

what you expect, the weather is what you get”. Using a variety of sources including 39

reanalyses and paleo data, and aided by notions and analysis techniques from 40

Nonlinear Geophysics, we argue that this dictum is fundamentally wrong. In 41

addition to the weather and climate, there is a qualitatively distinct intermediate 42

regime extending over a factor of ≈ 1000 in scale. For example, mean temperature 43

fluctuations increase up to about 5 K at 10 days (the lifetime of planetary 44

structures), then decrease to about 0.2 K at 30 years, and then increase again to 45

about 5 K at glacial-‐interglacial scales. 46

Both deterministic GCM’s with fixed forcings (“control runs”) and stochastic 47

turbulence-‐based models reproduce the first two regimes, but not the third. The 48

middle regime is thus a kind of “macroweather” not “high frequency climate”. 49

Averaging macroweather over periods increasing to ≈ 30 yrs yields apparently 50

converging values: macroweather is “what you expect”. Macroweather averages 51

over ≈ 30 years have the lowest variability, they yield well defined “climate states” 52

and justify the otherwise ad hoc “climate normal” period. However, moving to 53

longer periods, these states increasingly fluctuate: just as with the weather, the 54

climate changes in an apparently unstable manner; the climate is not what you 55

expect. Similarly, we may categorize climate forcings according to whether their 56

fluctuations decrease or increase with scale and this has important implications for 57

GCM’s and for climate change and climate predictions. 58

1. Introduction 59

In his monumental “Climate: Past, Present, and Future” Horace Lamb argued that 60

the early scientific view was “climate as constant” [Lamb, 1972]. Reflecting this, in 61

1935 the International Meteorological Organization adopted 1901-1930 as the “climatic 62

normal period”. Following the post war cooling, this view evolved: for example the 63

official American Meteorological Society glossary [Huschke, 1959] defined the climate 64

as “the synthesis of the weather” and then “…the climate of a specified area is 65

represented by the statistical collective of its weather conditions during a specified 66

interval of time (usually several decades)”. Although this new definition in principle 67

allows for climate change, the period 1931-1960 soon became the new “normal”, the ad 68

hoc 30 year duration became entrenched, today 1961-1990 is commonly used. Mindful 69

of the extremes, Lamb warned against reducing the climate to just “average weather”, 70

while viewing the climate as “…the sum total of the weather experienced at a place in the 71

course of the year and over the years”, [Lamb, 1972]. 72

Lamb’s essentially modern view allows for the possibility of climate change 73

and is closely captured by the popular expression: “The climate is what you expect, 74

the weather is what you get” (the character Lazurus Long in [Heinlein, 1973], but 75

often attributed to Mark Twain). It is also close to the US National Academy of 76

Science definition: “Climate is conventionally defined as the long-‐term statistics of 77

the weather…” [Committee on Radiative Forcing Effects on Climate, 2005] which 78

improves on the “the climate is what you expect” idea only a little by proposing: 79

“…to expand the definition of climate to encompass the oceanic and terrestrial 80

spheres as well as chemical components of the atmosphere”. 81

The Twain/Heinlein expression was strongly endorsed by the late E. Lorenz 82

who stated: “Before embarking on a search for an ideal definition (of climate) 83

assuming one exists, let me express my conviction that such a definition, when 84

found must agree in spirit with the statement, “climate is what you expect”.” [Lorenz, 85

1995]. He then proposed several definitions based on dynamical systems theory 86

and strange attractors (see also [Palmer, 2005]). 87

A variant on this, motivated by GCM modeling, was proposed by [Bryson, 1997] 88

(criticized by [Pielke, 1998]): “Climate is the thermodynamic/ hydrodynamic status of 89

the global boundary conditions that determine the concurrent array of weather patterns.” 90

He explains that whereas “weather forecasting is usually treated as an initial value 91

problem … climatology deals primarily with a boundary condition problem and the 92

patterns and climate devolving there from”. This definition could be paraphrased “for 93

given boundary conditions, the climate is what you expect”. This and similar views 94

provide the underpinnings for much of current climate prediction, including the recent 95

idea of “seamless forecasting” (e.g. [Palmer et al., 2008], [Palmer, 2012]) in which 96

seasonal scale model validation is applied to climate scale predictions (for a recent 97

discussion, see [Pielke et al., 2012]). 98

There are two basic problems with the Twain/Heinlein dictum and its variants. 99

The first is that they are based on an abstract weather - climate dichotomy, they are not 100

informed by empirical evidence. The glaring question of how long is “long” is either 101

decided subjectively or taken by fiat as the WMO’s “normal” 30 year period. The second 102

problem – that will be evident momentarily – is that it assumes that the climate is nothing 103

more than the long-term statistics of weather. If we accept the usual interpretation – that 104

we “expect” averages - then the dictum means that averaging weather over long enough 105

time periods converges to the climate. With regards to external forcings, one could argue 106

that this notion could still implicitly include the atmospheric response i.e. with averages 107

converging to slowly varying “responses”. However, it implausibly excludes the 108

appearance of any new “slow”, internal climate processes leading to fluctuations growing 109

rather than diminishing with scale. 110

To overcome this objection, one might adopt a more abstract interpretation of 111

what we “expect”. For example if the climate is defined as the probability distribution of 112

weather states, then all we “expect” is a random sample. However even this works only 113

inasmuch as it is possible to merge fast weather processes with slow climate processes 114

into a single process with a well defined probability distribution. While this may satisfy 115

the theoretician, it is unlikely to impress the layman. This is particularly true since to be 116

realistic we will see that one of the regimes in this composite model must have the 117

property that fluctuations converge whereas in the other, they diverge with scale. To use 118

the single term “climate” to encompass the two opposed regimes therefore seems at best 119

unhelpful and at worst misleadling. 120

The main purpose of this paper is show that the weather-climate dichotomy is 121

empirically untenable, that hiding between the two is a missing middle range spanning a 122

factor of a thousand in scale (≈ 10 days to ≈ 30 -100 years) characterized by qualitatively 123

different dynamics. This new low frequency weather regime was dubbed “macroweather” 124

since it is a kind of large scale weather (not small scale climate) regime [Lovejoy and 125

Schertzer, 2013], it fundamentally clarifies the distinction between weather and climate, 126

the status and role of GCM models and the notion of climate predictability. 127

2. The variability characterized by spectral composites 128

Notwithstanding the existence of several strong periodicities (notably 129

diurnal and annual), the atmosphere is highly variable over huge ranges of space-‐130

time scales. In addition, it has long been recognized (e.g. [Lovejoy and Schertzer, 131

1986], [Wunsch, 2003]) that even at the longer climate scales, the contribution to 132

the variability from specific frequencies associated with specific quasi periodic 133

processes is small: the great majority of the variance in the spectrum is from the 134

continuous background “noise”. Any objectively based definition of weather or 135

climate must therefore start from a clear picture of the atmosphere’s temporal 136

variability over wide scale ranges. 137

The first -‐ and still most ambitious -‐ single composite spectrum of 138

atmospheric variability [Mitchell, 1976], ranged from hours to the age of the earth (≈ 139

10-‐4 to 1010 yrs), fig. 1 shows a modern version. Given the rudimentary quality of the 140

data at that time, Mitchell admitted that his composite was mostly an “educated 141

guess” (indeed, over the range ≈ 104 to ≈ 108 yrs, [Shackleton and Imbrie, 1990] 142

empirically found that Mitchell had underestimated the variation of the spectral 143

density by a factor of ≈ 105!). His framework reflected the prevailing idea that there 144

were numerous roughly periodic processes, superposed onto a continuous 145

background spectrum E(ω) made up of a hierarchy of white noise processes and 146

their integrals (i.e. Ornstein-‐Uhlenbeck processes with spectra E(ω) ≈ ω-‐β with β = 0, 147

2 respectively where β is the spectral exponent: the negative slope on the log-‐log 148

plot in fig. 1). The spectral spikes were therefore superposed on a spectrum 149

consisting of a series of “shelves” and represented distinct physical processes. 150

Mitchell explained his idea as follows: 151

152

“As we scan the spectrum from the short-‐wave end toward the longer wave 153

regions, at each point where we pass through a region of the spectrum 154

corresponding to the time constant of a process that adds variance to the climate, 155

the amplitude of the spectrum increases by a constant increment across all 156

substantially longer wavelengths. In other words, each stochastic process adds a 157

shelf to the spectrum at an appropriate wavelength” [Mitchell, 1976]. 158

159

By the early 1980’s, following the explosion of scaling (fractal) ideas it was 160

realized that scale invariance was a very general symmetry principle often 161

respected by nonlinear dynamics, including many geophysical processes and 162

turbulence. The signature of a scaling process is a power law spectrum, linear on a 163

log-‐log plot. Although in order to accommodate the wide range of scales, Mitchell 164

had found it “necessary to resort to logarithmic coordinates”, there was no 165

implication that the underlying processes might have nontrivial scaling over any 166

significant range. In contrast, scaling symmetries, were explicitly invoked to justify 167

the alternative composite picture ([Lovejoy and Schertzer, 1984; Lovejoy and 168

Schertzer, 1986] which profited from early ocean and ice core paleotemperatures. 169

These analyses already clarified the following points: a) the distinction between the 170

variability of regional and global scale temperatures, b) that there was a scaling 171

range for global averages between scales of the order of 10 yrs (τc in the notation 172

here) up to ≈ 40 -‐ 50 kyrs with an exponent βc ≈ 1.8, c) that a scaling regime with this 173

exponent could quantitatively explain the magnitudes of the temperature swings 174

between interglacials (“ice ages”): the “interglacial window” (see below). 175

In the last 15 years this picture has been supported by the quite similar 176

scaling composites of [Pelletier, 1998] and [Huybers and Curry, 2006]. The latter in 177

particular made a data intensive study of the scaling of many different types of 178

paleotemperatures collectively spanning the range of about 1 month to nearly 106 179

years. In addition, even without producing composites, other authors shared the 180

scaling framework, e.g. [Koscielny-Bunde et al., 1998], [Talkner and Weber, 2000], 181

[Ashkenazy et al., 2003; Rybski et al., 2008]. Their results are qualitatively very 182

similar -‐ including the positions of the scale breaks; the main innovations are a) the 183

increased precision on the β estimates and b) the basic distinction made between 184

continental and oceanic spectra including their exponents. We could also mention 185

the composite of [Fraedrich et al., 2009] which is a modest adaptation of Mitchell’s: 186

it innovates by introducing a single scaling regime from ≈ 3 to ≈ 100 yrs. 187

Using real temperature and paleotemperature data, examples showing the 188

behaviors in the three different regimes are graphically illustrated in fig. 2. Notice 189

that in the weather regime (bottom) the temperature seems to “wander” up or down 190

like a drunkard’s walk so that temperature differences typically increase over longer 191

and longer periods. Turning to the middle macroweather series, we see that it has a 192

totally different appearance with successive fluctuations on the contrary tending to 193

cancel each other out, i.e. with decreases followed by partially cancelling increases 194

(and visa versa). At first sight, this vindicates the “climate is what you get” idea 195

since averages over longer and longer periods will clearly converge. From this, we 196

anticipate that at decadal -‐ or certainly at centennial scales -‐ that we will see at most 197

smooth, slow variations. However, when we turn to the century resolution climate 198

series (top) on the contrary, we once again see weather-‐like wandering. 199

Below we shall see that the fluctuations over a time interval Δt are in each 200

case roughly scaling (power laws) of the form ΔtH so that the sign of H qualitatively 201

distinguishes the “wandering” (H>0) or “cancelling” (H

Starting with the climate regime, numerous paleo temperature series (mostly from ice 213

and ocean cores) have been analyzed and there is broad agreement on their scaling nature 214

with spectral exponents estimated in the range βc ≈ 1.3 to 2.1 over range from hundreds to 215

tens of thousands of years, [Lovejoy and Schertzer, 1986], [Schmitt et al., 1995], 216

[Ditlevsen et al., 1996], [Pelletier, 1998], [Ashkenazy et al., 2003], [Wunsch, 2003], 217

[Huybers and Curry, 2006], [Blender et al., 2006] , [Lovejoy and Schertzer, 2012c]. 218

These analyses employed diverse techniques including spectra, difference and Haar 219

structure functions as well as Detrended Fluctuation Analysis so that the results are fairly 220

robust. In addition, as discussed below (fig. 3), further analyses from surface 221

temperatures, multiproxy reconstructions and 138 year long Twentieth Century reanalysis 222

(20CR), lend this further quantitative support. 223

Similarly, in the macroweather regime, there are now many studies finding 224

scaling with spectral exponents βmw

Of the three regimes, the only one where the idea of a roughly scaling 236

background spectrum is still somewhat controversial is the higher frequency 237

weather regime (scales < τw ≈ 10 days). To understand the debate, recall that the 238

classical turbulence theories describing the statistical variability in the weather 239

regime are all based on isotropic scaling, the most famous being the Kolmogorov k-‐240

5/3 spectrum for the wind (k is a wavenumber). However, the strong vertical 241

atmospheric stratification prevents isotropic scaling from holding over any scale 242

ranges spanning the scale thickness of the atmosphere ( ≈ 10 km). One must 243

therefore abandon either the scaling or the isotropy assumption. Following 244

Kraichnan’s development of 2-‐D turbulence and Charney’s extension to (still 245

essentially 2D) “quasi geostrophic” turbulence, the usual choice was to retain 246

isotropy and to divide the dynamics into 2D isotropic (large scale) and 3D isotropic 247

(small) scale regimes ([Kraichnan, 1967], [Charney, 1971]). However, starting with 248

[Schertzer and Lovejoy, 1985], a growing body of evidence and theory has 249

supported the alterative anisotropic scaling hypothesis. Thanks both to modern 250

empirical evidence (e.g. the review [Lovejoy and Schertzer, 2010], [Lovejoy and 251

Schertzer, 2013] and a recent massive aircraft study [Pinel et al., 2012]), but also to 252

theoretical arguments showing that the governing equations are symmetric with 253

respect to anisotropic scaling symmetries ([Schertzer et al., 2012]), the question 254

increasingly has been settled in favor of anisotropic scaling (see the recent debate 255

[Lovejoy et al., 2009], [Lindborg et al., 2010a; Lindborg et al., 2010b], [Lovejoy et 256

al., 2010], [Schertzer et al., 2011], [Yano, 2009], ([Schertzer et al., 2012]). The 257

implications of this anisotropic spatial scaling for the temporal statistics are 258

discussed in [Radkevitch et al., 2008] and [Lovejoy and Schertzer, 2010]. 259

A review of diverse evidence from reanalyses, in situ and remotely sensed 260

data ([Lovejoy and Schertzer, 2010], [Lovejoy and Schertzer, 2013]) shows that 261

for wind, temperature, humidity, pressure, short and long wave radiances, βw is 262

commonly in the range 1.5 -‐2 (certainly >1, hence H>0). The existence of a basic 263

transition in the range ≈ 5 – 20 days has been recognized at least since [Van der 264

Hoven, 1957] who noted a low frequency spectral “bump” at around 5 days. Later, 265

the corresponding features in the temperature and pressure spectra were termed 266

“synoptic maxima” by [Kolesnikov and Monin, 1965] and [Panofsky, 1969]. More 267

recently, in the same spirit as Mitchell, the transition has been modeled (e.g. 268

[AchutaRao and Sperber, 2006]) as an Orenstein-‐Uhlenbeck process i.e. with βw = 2, 269

βmw = 0, corresponding to Hw = 1/2, Hmw = -‐1/2, although as can be seen in fig. 3 270

(discussed below), this is not a very accurate approximation and can be misleading. 271

Finally, [Vallis, 2010] proposed a (nonscaling) mechanism by suggesting that τw is 272

determined by the lifetimes of baroclinic instabilities. These were estimating by the 273

inverse Eady growth rate (τEady) which yielded τEady ≈ 4 days. However this result 274

requires a linearization of the equations about a hypothetical state having uniform shear 275

and uniform stratification across the entire troposphere. In comparison, the real 276

troposphere has highly nonuniform shears and stratifications, its variability is so 277

strong that it is characterized by anomalous scaling exponents throughout [Lovejoy 278

et al., 2007] (including H ≈ 0.75 for the horizontal wind in the vertical direction). 279

Another difficulty with using τEady ≈ τw is that τEady is inversely proportional to the 280

Coriolis parameter so that it diverges at the equator whereas the empirical τw is not very 281

sensitive to latitude. 282

Indeed, a seductive feature of the (anisotropic) scaling framework is that it 283

fairly accurately predicts the weather to macroweather transition scale τw ≈ 10 days. 284

The argument is as follows: the sun provides ≈ 240 W/m2 of heating with a 2% 285

efficiency of conversion to kinetic energy ([Monin, 1972]). The energy is distributed 286

reasonably uniformly over the troposphere, leading to a turbulent energy flux 287

density (ε) close to the observed global value ε ≈ 10-‐3 W/kg ([Lovejoy and Schertzer, 288

2010]; this is the flux of energy from large to small scales). The model predicts that 289

this turbulent energy flux is the fundamental driver of the horizontal dynamics and 290

thus that planetary structures have eddy-‐turnover times of ≈ ε-‐1/3Le2/3 ≈ 10 days 291

where Le =20000 km is the largest great circle distance on the earth. The analogous 292

calculation for the ocean using the empirical ocean turbulent flux ε ≈ 10-‐8 W/kg, 293

yields a lifetime of ≈ 1 yr which is indeed the scale separating a high frequency 294

“ocean weather” (with β >1) from a low frequency “macro-‐ocean weather” with β

structures with lifetimes Δt; at scales Δt>τw they depend on the statistics of many 304

planetary sized structures. In addition, the basic FIF model predicts [Lovejoy and 305

Schertzer, 2012c] macroweather exponents typically in the range 0.2 < βmw < 0.6 306

(i.e. -‐0.4 < Hmw < -‐0.2). 307

4. Real space fluctuations and analyses 308

In spite of the now burgeoning evidence that the atmosphere’s natural 309

variability is scaling over various ranges, the idea has not received the attention it 310

deserves and at least over decadal, centennial and millennial scales, the natural 311

variability is still largely identified with quasi-‐periodic behaviours which – when 312

present -‐ have the advantage of being predictable (for examples of periodicities 313

ranging from multidecadal to millennial scales see [Delworth et al., 1993], 314

[Schlesinger and Ramankutty, 1994], [Mann and Park, 1994], [Mann et al., 1995], 315

[Bond et al., 1997], [Isono et al., 2009]). An additional reason for this focus on 316

quasi-‐periodic behavior is that whereas spectra are ideal for understanding periodic 317

processes, they are not optimal for scaling processes. For these, the corresponding 318

real space analyses are more straightforward to interpret; this is particularly true 319

when comparing spectra from different data types with different resolutions. In 320

this section we show how this works. 321

In order to understand the qualitatively different behaviors in fig. 2, consider 322

fluctuations ΔT. In a scaling regime, these will change with scale as ΔT = ϕ ΔtH where 323

H is the fluctuation exponent (also called the “nonconservation” exponent; it is denoted 324

“H” in honor of Edwin Hurst but in general – unless the process is Gaussian - it is not the 325

same as the Hurst exponent). ϕ is a controlling dynamical variable (e.g. a turbulent flux) 326

whose mean < ϕ > is independent of the lag Δt (i.e. independent of the time scale). The 327

behavior of the mean fluctuation is thus ≈ ΔtH so that if H>0, on average 328

fluctuations tend to grow with scale whereas if H0), mean 335

absolute differences cannot decrease and so when H

Once estimated, the variation of the fluctuations with scale can be quantified 347

by using their statistics; the qth order structure function Sq(Δt) is particularly 348

convenient: 349

Sq Δt( ) = ΔT Δt( )q (1) 350

where “” indicates ensemble averaging. In a scaling regime, Sq(Δt) is a power 351

law; Sq(Δt) ≈ Δtξ(q), where the exponent ξ(q) = qH – K(q) where K(1) = 0. Since 352

Gaussian processes have K(q) = 0, K(q) characterizes the strong non Gaussian 353

variability; the “intermittency”. In the macroweather regime K(2) is typically small 354

(≈0.01-‐0.03), so that the RMS variation S2(Δt)1/2 (denoted simply S(Δt) below) has 355

the exponent ξ(2)/2 ≈ ξ(1) = H. In the climate regime this intermittency correction 356

is a bit larger [Schmitt et al., 1995] but the error in using this approximation ( ≈0.06) 357

will be neglected. Note that since the spectrum is a second order statistic, we have 358

the useful relationship β = 1+ξ(2) = 1+2H-K(2). When K(2) is small, β ≈ 1+2H so that 359

as mentioned earlier, H>0, H 1, β0, Hmw0). The transitions are at τw ≈ 5 -‐ 10 365

days and τc ≈ 30 -‐100 yrs: note that in the recent period τc is a bit lower (≈ 10 yrs) -‐ 366

see the 1880-‐2008 surface series – than in earlier periods (see the multiproxies with 367

τc ≈ 30 -‐100 yrs), this is because over the last century, anthropogenic forcing has 368

become dominant for scales greater than about 10 yrs, see [Lovejoy, 2013]. We can 369

also glimpse a fourth low frequency “macro climate” regime for scales larger than 370

τmc ≈ 100 kyrs, but this is outside our scope. b) the difference between the local and 371

global fluctuations. c) the “glacial/interglacial window” corresponding to overall ±3 372

to ±5 K variations (i.e. S(Δt) ≈ 6, 10 K) over scales with half periods of 30 – 50 kyrs; 373

the curve must pass through the window in order to explain the glacial/interglacial 374

transitions. Starting at τc ≈ 10 -‐ 30 yrs, one can plausibly extrapolate the global 375

surface and 20CR 700 mb S(Δt)’s using H = 0.4 (β ≈1.8), all the way to the 376

interglacial window (with nearly an identical S as in [Lovejoy and Schertzer, 1986]). 377

Similarly, the local temperatures and multiproxies also seem to follow the same 378

exponent with slightly different τc’s and seem to extrapolate respectively a little 379

above and below the window. 380

These statistics may seem arcane but their physical interpretation is pretty 381

straightforward. In the weather regime, larger and larger fluctuations “live” for 382

longer and longer times: the “eddy turnover time”. At any given time scale, the 383

fluctuations are dominated by structures with corresponding spatial scales and this 384

relationship holds up to structures of planetary scales with lifetimes ≈ 10 days. For 385

periods longer than this, the statistics are dominated by averages of many planetary 386

scale structures, and these fluctuations tend to cancel out: for example large 387

temperature increases are typically followed (and partially cancelled) by 388

corresponding decreases. The consequence is that in this macroweather regime, the 389

average fluctuations diminish as the time scale increases. At some point – at around 390

30 years depending on geographic location and time (and as little as 10 years in the 391

recent period when anthropogenic variability is important), these weaker and 392

weaker fluctuations -‐ whose origin is in weather dynamics -‐ become dominated by 393

increasingly strong lower frequency processes. These not only include changing 394

external solar, volcanic orbital or anthropogenic “forcings” – but quite likely also 395

new and increasingly strong slow (internal) climate processes -‐ or by a combination 396

of the two: forcings with feedbacks. A relevant example of a slow dynamical process 397

that is not currently fully incorporated into GCM’s is land-‐ice dynamics. The overall 398

effect is that in the resulting climate regime, fluctuations tend to grow again with 399

scale in an “unstable” manner, very similar to the way they grow in the weather 400

regime. 401

5. Implications for climate modelling, prediction 402

Numerical weather models and reanalyses are qualitatively in good agreement 403

with the weather / macroweather picture described above, although there are still 404

some quantitative discrepancies in the values of the exponents, possibly due the 405

hydrostatic approximation and other numerical issues ([Stolle et al., 2009], 406

[Lovejoy and Schertzer, 2011]). However, climate models (GCM’s) are essentially 407

weather models with various additional couplings (with ocean, carbon cycle, land-‐408

use, sea ice and other models). It is therefore not surprising that control runs (i.e. 409

with no “climate forcings”) generate macroweather (with βmw ≈ 0.2, Hmw ≈ -‐0.4), and 410

this apparently out to the extreme low frequency limit of the models (see the review 411

and analyses in [Lovejoy et al., 2012] as well as [Blender et al., 2006], [Rybski et al., 412

2008]). 413

Avoiding anthropogenic effects by considering the pre-‐1900 epoch, for GCM 414

climate models, the key question is whether solar, volcanic, orbital or other climate 415

forcings are sufficient to arrest the H 0 regime with sufficiently strong centennial, millennial variability to 417

account for the background variability out to glacial-‐ interglacial scales. Analysis of 418

several last millennium simulations has found that at the moment, their low 419

frequency variabilities are too weak [Lovejoy et al., 2012]. 420

To understand this weak variability, one can examine the scale dependence of 421

fluctuations in the radiative forcings (ΔRF) of several solar and volcanic 422

reconstructions; they are generally scaling with ΔRF ≈ AΔtHR [Lovejoy and Schertzer, 423

2012a]. If HR ≈HT ≈ 0.4, then scale independent amplification / feedback mechanisms 424

would suffice. However it was often found that HR ≈ -‐0.3 implying that the forcings 425

become weaker with scale -‐ even though the response grows with scale. This 426

suggests the need to introduce new slow mechanisms of internal variability. Such 427

mechanisms must have broad spectra; this suggests that their dynamics involve 428

nonlinearly interacting spatial degrees of freedom such as the land-‐ice dynamics 429

mentioned above. 430

Whatever the ultimate source of the growing fluctuations in the H > 0 climate 431

regime, a careful and complete characterization of the scaling in space as well as in 432

time (including possible space-‐time anisotropies) allows for new stochastic methods 433

for predicting the climate. The idea is to exploit the particularly low variability of 434

the averages at scale τc. Since τc ≈ 30 yrs -‐ i.e. the conventional but ad hoc “climate 435

normal” period -‐ this not only justifies the normal but allows averages of relevant 436

variables over it to define “climate states” and the changes at scales Δt>τc to define 437

climate change (again, in the recent period, this defines the scale at which 438

anthropogenic variability starts to dominate natural variability). Even without 439

resolving the question of the dominant climate forcing and slow internal feedbacks, 440

one could use the statistical properties of the climate states -‐ the system’s “memory” 441

implicit in the long range statistical correlations – combined with the growing data 442

on past climate states in order to make stochastic climate forecasts. 443

Another attractive application of this scaling picture is that by quantifying the 444

natural variability as a function of space-‐time scales, it opens up the possibility of 445

convincingly distinguish natural and anthropogenic variability. This is possible 446

because the stochastic scaling framework allows one to statistically test specific 447

hypotheses about the probability that the atmosphere would naturally behave in the 448

way that is observed, i.e. to formulate rigorous statistical tests of any trends or 449

events against the null hypothesis. Only if the probabilities are low enough should 450

the hypothesis that the observed changes are natural in origin be rejected. For 451

example, using CO2 as a surrogate for all anthropogenic effects and taking into both 452

long range statistical dependencies and extreme “fat tailed” probabilities, [Lovejoy, 453

2013] found that the probability of the global warming since 1880 being due to 454

natural variability can be rejected with more than 99% confidence. The scaling 455

flucutation approach thus allows quantitative (and hence convincing) answers to 456

questions such as: how can the earth have prolonged periods of cooling in the midst 457

of anthropogenic warming; or was this winter’s record mild temperature really 458

evidence for anthropogenic influence? Finally, the systematic comparison of model 459

and natural variability in the preindustrial era is the best way to fully address the 460

issue of “model uncertainty”, to assess the extent by which the models are missing 461

important slow processes. 462

6. Conclusions 463

Contrary to [Bryson, 1997], we have argued that the climate is not accurately 464

viewed as the statistics of fundamentally fast weather dynamics that are constrained by 465

quasi fixed boundary conditions. The empirically substantiated picture is rather one of 466

“unstable”, “wandering”, high frequency weather processes (i.e. H>0) tending - at scales 467

beyond 10 days or so and primarily due to the quenching of spatial degrees of freedom 468

(intermediate frequency, low variability) to macroweather processes. These appear to be 469

stable because positive and negative fluctuations tend to cancel out (H0) and only emerge from 471

macroweather at even lower frequencies, due to new slow internal climate processes 472

coupled with external forcings (including in the recent period, anthropogenic forcings). 473

These processes presumably include various nonlinear couplings with the fields that 474

Bryson considered to be no more than “boundary conditions” so that “rather than 475

‘boundaries’ these become interactive mediums” [Pielke, 1998]. The 476

theoretical/mathematical picture underpinning GCM based prediction beyond ≈ 30 years 477

should thus be re-examined. 478

Yet, whatever the cause, it is an empirical fact that the emergent synergy of new 479

processes yields fluctuations that on average again grow with scale in at least a roughly 480

scaling manner and become dominant typically on time scales of 30 -100 years 481

(somewhat less in the recent period) up to ≈ 100 kyrs. 482

Looked at another way, if the climate really was what you expected, then – 483

since one usually expects averages -‐ predicting the climate would be the relatively 484

simple matter of determining the fixed climate normal. On the contrary, we have 485

argued that from the stochastic point of view -‐ and notwithstanding the vastly 486

different time scales -‐ that predicting natural climate change is very much like 487

predicting the weather. This is because the climate at any time or place is the 488

consequence of climate changes that are (qualitatively and quantitatively) 489

unexpected in very much the same way that the weather is unexpected. 490

At a subjective level, from experience over the years, we all grow to expect 491

certain stable patterns of macroweather (complicated by seasonal, and now by 492

anthropogenic effects, but nevertheless recognizable from year to year) so that in 493

daily discourse we may say “macroweather is what you expect, the weather is what 494

you get”. However over generational scales -‐ periods of 30 years -‐ the 495

macroweather we are accustomed to evolves due to climate change. Speaking to our 496

children and grandchildren, the appropriate dictum would therefore be 497

“macroweather is what you expect, the climate is what you get”. 498

6. Acknowledgements: 499 We would like to thank Roger Pielke, Gavin Schmidt, and Adrian Tuck for 500

useful comments. 501

502

7. References 503

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687 688

689

Figures and Captions: 690

691

692

Fig. 1: A composite temperature spectrum. Left: the GRIP (Summit) ice core δ18O (a 693

temperature proxy, low resolution (55 cm in depth back 240 kyrs) along with the first 91 694

kyrs at high resolution. Right: the spectrum of the (mean) 75oN of the 20th Century 695

Reanalysis (20CR) temperature spectrum, at 6 hour resolution, from 1871-2008, at 700 696

mb. The overlap (from 10 – 138 yr scales) is used for calibrating the former (moving 697

them vertically on the log-log plot). All spectra are averaged over logarithmically spaced 698

bins, ten per order of magnitude in frequency. Three regimes are shown corresponding to 699

the weather regime with βw= 2 (the diurnal variation and subharmonic at 12 hours are 700

visible at the extreme right). The central macroweather “plateau” is shown along with the 701

theoretically predicted βmw = 0.2 - 0.4 regime. Finally, at longer time scales (the left), a 702

MacroweatherClimate Weather

m

new scaling climate regime with exponent βc ≈1.4 continues to about 100 kyrs. 703

Reproduced from [Lovejoy and Schertzer, 2012c]. 704

705

706

Fig. 2: Dynamics and types of scaling variability: A visual comparison displaying 707

representative temperature series from weather, macroweather and climate (H ≈ 708

0.4, -‐0.4, 0.4, bottom to top respectively). To make the comparison as fair as possible, 709

in each case, the sample is 720 points long and each series has its mean removed 710

and is normalized by its standard deviation (4.49 K, 2.59 K, 1.39 K, respectively), the 711

two upper series have been displaced in the vertical by four units for clarity. The 712

resolutions are 1 hour, 20 days and 100 yrs, the data are from a weather station in 713

Lander Wyoming (detrended daily, annually), the 20th Century reanalysis and the 714

100 200 300 400 500 600 700

- 2

2

4

6

8

10

1 Century,Vostok,20-92kyr BP

20 days,75oN,100 oW,1967 - 2008

1 hour,Lander, 10 Feb.-12 March, 2005

0

T/

t

Vostok Antarctic station. Note the similarity between the type of variability in the 715

weather and climate regimes (reflected in their scaling exponents although the H 716

exponent is only a partial characterization). 717

718

719

Fig. 3: Empirical RMS temperature fluctuations (S(Δt)), local scale analyses: On 720

the left top we show grid point scale (2ox2o) daily scale fluctuations globally 721

averaged along with reference slope ξ(2)/2 = -‐0.4 ≈ H (20CR, 700 mb). For 722

comparison, the results for 50 simulations of Orenstein-Uhlenbeck (OU) processes are 723

also given using simulations with a characteristic time of 3 days. The theoretical 724

asymptotic slopes (0.5, -0.5) are added to show their convergence to theory. Just below 725

this, we show the monthly NOAA CDC Sea Surface Temperature curve (5o resolution, 726

from 1900-2000); the transition from ξ(2)/2 ≈ 0.4 to ≈ -‐0.2 occurs at τow ≈ 1 yr. On the 727

lower left, we see at daily resolution, the corresponding globally averaged structure 728

function. 729

Globally averaged series: The same 20CR data but globally averaged (brown), The 730

0.5

0.5- 0

average of the three in situ surface series (NOAA NCDC, NASA GISS, CRU, red) as well 731

as the average of three post 2003 multiproxy structure functions; [Huang, 2004], 732

[Ljundqvist, 2010; Moberg et al., 2005], (see [Lovejoy and Schertzer, 2012c]). 733

Paleotemperatures: At the right we show analysis of the EPICA Antarctic series 734

interpolated to 50 year resolution over ≈ 800 kyrs. Also shown is the interglacial 735

“window”. 736

The reference slopes are ξ(2)/2 = -0.4, -0.2 or +0.4; these correspond to spectral 737

exponents β = 1+ξ(2) = 0.2, 0.6, 1.8 respectively; a flat curve in the above corresponds to 738

β = 1. 739

740

Figure Captions: 741

Fig. 1: A composite temperature spectrum. Left: the GRIP (Summit) ice core δ18O (a 742

temperature proxy, low resolution (55 cm in depth back 240 kyrs) along with the first 91 743

kyrs at high resolution. Right: the spectrum of the (mean) 75oN of the 20th Century 744

Reanalysis (20CR) temperature spectrum, at 6 hour resolution, from 1871-2008, at 700 745

mb. The overlap (from 10 – 138 yr scales) is used for calibrating the former (moving 746

them vertically on the log-log plot). All spectra are averaged over logarithmically spaced 747

bins, ten per order of magnitude in frequency. Three regimes are shown corresponding to 748

the weather regime with βw= 2 (the diurnal variation and subharmonic at 12 hours are 749

visible at the extreme right). The central macroweather “plateau” is shown along with the 750

theoretically predicted βmw = 0.2 - 0.4 regime. Finally, at longer time scales (the left), a 751

new scaling climate regime with exponent βc ≈1.4 continues to about 100 kyrs. 752

Reproduced from [Lovejoy and Schertzer, 2012c]. 753

Fig. 2: Dynamics and types of scaling variability: A visual comparison displaying 754

representative temperature series from weather, macroweather and climate (H ≈ 755

0.4, -‐0.4, 0.4, bottom to top respectively). To make the comparison as fair as possible, 756

in each case, the sample is 720 points long and each series has its mean removed 757

and is normalized by its standard deviation (4.49 K, 2.59 K, 1.39 K, respectively), the 758

two upper series have been displaced in the vertical by four units for clarity. The 759

resolutions are 1 hour, 20 days and 100 yrs, the data are from a weather station in 760

Lander Wyoming (detrended daily, annually), the 20th Century reanalysis and the 761

Vostok Antarctic station. Note the similarity between the type of variability in the 762

weather and climate regimes (reflected in their scaling exponents although the H 763

exponent is only a partial characterization). 764

Fig. 3: Empirical RMS temperature fluctuations (S(Δt)), local scale analyses: On 765

the left top we show grid point scale (2ox2o) daily scale fluctuations globally 766

averaged along with reference slope ξ(2)/2 = -‐0.4 ≈ H (20CR, 700 mb). For 767

comparison, the results for 50 simulations of Orenstein-Uhlenbeck (OU) processes are 768

also given using simulations with a characteristic time of 3 days. The theoretical 769

asymptotic slopes (0.5, -0.5) are added to show their convergence to theory. Just below 770

this, we show the monthly NOAA CDC Sea Surface Temperature curve (5o resolution, 771

from 1900-2000); the transition from ξ(2)/2 ≈ 0.4 to ≈ -‐0.2 occurs at τow ≈ 1 yr. On the 772

lower left, we see at daily resolution, the corresponding globally averaged structure 773

function. 774

Globally averaged series: The same 20CR data but globally averaged (brown), The 775

average of the three in situ surface series (NOAA NCDC, NASA GISS, CRU, red) as well 776

as the average of three post 2003 multiproxy structure functions; [Huang, 2004], 777

[Ljundqvist, 2010; Moberg et al., 2005], (see [Lovejoy and Schertzer, 2012c]). 778

Paleotemperatures: At the right we show analysis of the EPICA Antarctic series 779

interpolated to 50 year resolution over ≈ 800 kyrs. Also shown is the interglacial 780

“window”. 781

The reference slopes are ξ(2)/2 = -0.4, -0.2 or +0.4; these correspond to spectral 782

exponents β = 1+ξ(2) = 0.2, 0.6, 1.8 respectively; a flat curve in the above corresponds to 783

β = 1. 784

785

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